HEXCO · Polytechnic · NC Accountancy

Financial Mathematics and Statistics 1: Financial Statistics

Learning Outcome 2 of 3

Free HEXCO NC Accountancy notes on Financial Statistics, learning outcome 2 of 3 in Financial Mathematics and Statistics 1. Business computations, financial statistics and the time value of money. Semester 2, module 310/22/M05.

Here we learn the meaning, purposes, uses and limitations of statistics, how data is collected, sampled, classified, tabulated and presented, and how to calculate and interpret the mean, median, mode, range, variance, standard deviation and coefficient of variation.

Managers are surrounded by figures on sales, costs, wages and prices, and statistics gives them methods to collect, summarise and understand those figures before making decisions.

2.1 Introduction to Statistics

The word statistics is used in two ways: for numerical facts themselves, and for the methods used to handle them.

As a subject, it gives business a disciplined way of turning raw numbers into useful information.

Meaning of Statistics

In the plural sense, statistics are numerical facts collected for a purpose, such as monthly sales figures or inflation rates.

In the singular sense, statistics is the science of collecting, organising, presenting, analysing and interpreting numerical data in order to draw conclusions and make decisions.

Purposes and Uses of Statistics

Statistics simplifies large masses of data into averages, tables and graphs, allows comparison between periods, branches or products, shows trends that help with forecasting, and supports decision making with evidence. In business it is used for sales forecasting, market research, quality control, stock control, setting wages and prices, budgeting, and analysing financial performance. Government uses it for planning, for example through the census and price indices.

Limitations of Statistics

Statistics deals with groups, not individuals, so an average may not describe any one person. It ignores qualitative factors such as morale or customer loyalty. Results are only as good as the data, so errors or bias in collection lead to wrong conclusions. Statistics can be misused or misinterpreted to support a particular view, and conclusions from samples are true only on average and within a margin of error.

2.2 Collection of Data

Before any analysis, data must be collected in a way that suits the purpose and is reliable.

The choice of data type, collection method and sample all affect the quality of the results.

Types of Data

Primary data is collected first-hand by the investigator for a specific purpose, such as a customer survey; it is relevant and reliable but costly and slow. Secondary data has already been collected by someone else, such as ZimStat reports or company records; it is cheap and quick but may be out of date or not exactly suited. Data may also be quantitative (numerical), either discrete (counted, such as the number of employees) or continuous (measured, such as weight or time), or qualitative (descriptive categories, such as gender or department).

Methods of Data Collection

The main methods are questionnaires, sent or handed out for people to complete; interviews, face to face or by phone, which allow follow-up questions; observation, watching and recording what happens, such as counting customers; experiments, testing under controlled conditions; and records and documents, such as invoices, payroll and sales reports. Online forms and mobile surveys are now widely used because they are fast and cheap.

Sampling Methods

A population is the whole group being studied, and a sample is the part selected to represent it, because studying everyone is often too costly or slow.

MethodHow it works
Simple random sampling Every member has an equal chance.Names are drawn from a hat or chosen with random numbers.
Systematic sampling Every kth member.For example, every 10th invoice on a list, after a random start.
Stratified sampling Random samples from each group.The population is divided into groups, such as departments, and a proportional random sample is taken from each.
Cluster sampling Whole groups are chosen.Some branches are chosen at random, and everyone in those branches is studied.
Quota sampling A set number from each group.The interviewer fills a quota, such as 20 men and 20 women, choosing people conveniently.
Convenience sampling Whoever is easy to reach.Quick and cheap, but likely to be biased.

2.3 Classification and Tabulation of Data

Raw data collected in a survey is just a long list of numbers, which is hard to understand.

Classifying and tabulating it arranges the data so that patterns can be seen.

Meaning of Classification

Classification is the arranging of data into groups or classes according to common characteristics, such as by area (geographical), time (chronological), attributes (qualitative, such as department) or size (quantitative, such as wage bands).

It reduces a mass of data to a few groups, making comparison and further analysis possible.

Frequency Distribution Table

A frequency distribution shows how many times each value or class of values occurs. For grouped data, the range is divided into classes of equal width, each item is tallied into its class, and the tallies are counted to give the frequency (f). The class midpoint (x) is used in calculations. The table below shows the daily sales of 40 days at a small shop, and is used for the calculations in this outcome.

Tabulation of Data

Tabulation is the presentation of classified data in rows and columns. A good table has a title, a source note, clear column and row headings with units, totals, and is simple enough to read at a glance. Tables allow exact figures to be read and compared, and they are the starting point for graphs and calculations.

2.4 Presentation of Data

Charts and graphs show the shape of data far more quickly than a table.

The method chosen depends on the type of data and the message to be shown.

Methods of Presenting Data

The main methods are listed below. Every chart needs a title, labelled axes with units, a key where needed and a suitable scale.

Here you are working out the angles for a pie chart. Add up all the values to get the total, then for each item divide its value by the total and multiply by 360°. Check that the angles add up to 360° before you draw. Below, we show you how it is done, step by step.

MethodBest used for
Simple bar chart Bars of equal width, length showing the value.Comparing totals between categories, such as sales by branch.
Component and multiple bar charts Bars divided into parts, or groups of bars side by side.Showing the make-up of totals, or comparing several items over time.
Pie chart A circle divided into sectors, each angle = value ÷ total × 360°.Showing parts of a whole, such as the share of each expense.
Line graph Points joined by lines over time.Showing trends, such as monthly sales over a year.
Histogram Touching bars, area showing frequency.Showing a grouped frequency distribution of continuous data.
Frequency polygon and ogive Lines through midpoints, or through cumulative frequencies.Comparing distributions, and estimating the median and quartiles.
Pictogram Symbols representing quantities.Simple displays for the public.
Expenses for a month were: wages $3,000, rent $1,500, transport $1,000 and other $500. Find the angle for each sector of a pie chart.
  1. Total = 3,000 + 1,500 + 1,000 + 500 = $6,000
  2. Wages = 3,000 ÷ 6,000 × 360° = 180°
  3. Rent = 1,500 ÷ 6,000 × 360° = 90°
  4. Transport = 1,000 ÷ 6,000 × 360° = 60°
  5. Other = 500 ÷ 6,000 × 360° = 30°

Answer: 180°, 90°, 60° and 30°, which add up to 360°

A company's sales by branch are: Harare 45%, Bulawayo 30%, Mutare 15% and Gweru 10%. Find the pie chart angles.
  1. Harare = 45% × 360° = 162°
  2. Bulawayo = 30% × 360° = 108°
  3. Mutare = 15% × 360° = 54°
  4. Gweru = 10% × 360° = 36°
  5. Check: 162 + 108 + 54 + 36 = 360°

Answer: 162°, 108°, 54° and 36°

120 customers paid by: cash 30, EcoCash 60, card 20, bank transfer 10. Find the pie chart angles.
  1. Cash = 30 ÷ 120 × 360° = 90°
  2. EcoCash = 60 ÷ 120 × 360° = 180°
  3. Card = 20 ÷ 120 × 360° = 60°
  4. Bank transfer = 10 ÷ 120 × 360° = 30°

Answer: 90°, 180°, 60° and 30°

2.5 Measures of Central Tendency

A measure of central tendency is a single value that represents the centre, or typical value, of a set of data.

The three main measures are the mean, the median and the mode, and each has its own uses.

Meaning and Types

The arithmetic mean is the sum of the values divided by the number of values, x̄ = Σx ÷ n, or for a frequency distribution, x̄ = Σfx ÷ Σf. The median is the middle value when the data is arranged in order. The mode is the value that occurs most often.

Here you are finding one figure that sums up a whole set of data. For the mean, add all the values and divide by how many there are (for a table, use Σfx ÷ Σf). For the median, put the values in order and take the middle one, or the average of the two middle ones; the mode is simply the value that appears most often. Below, we show you how it is done, step by step.

Ungrouped: the weekly wages of 7 workers are $80, $95, $70, $95, $110, $85 and $95. Find the mean, median and mode.
  1. Mean = (80 + 95 + 70 + 95 + 110 + 85 + 95) ÷ 7 = 630 ÷ 7 = $90
  2. In order: 70, 80, 85, 95, 95, 95, 110; the middle (4th) value is $95
  3. The value that occurs most often is $95

Answer: Mean $90; median $95; mode $95

Ungrouped, even number of values: the units sold on 8 days were 4, 8, 6, 10, 12, 8, 15 and 9. Find the mean, median and mode.
  1. Mean = (4 + 8 + 6 + 10 + 12 + 8 + 15 + 9) ÷ 8 = 72 ÷ 8 = 9 units
  2. In order: 4, 6, 8, 8, 9, 10, 12, 15; with 8 values there are two middle values, the 4th and 5th
  3. Median = (8 + 9) ÷ 2 = 8.5 units
  4. Mode = 8 units, which appears twice

Answer: Mean 9; median 8.5; mode 8

Grouped: find the mean, median and mode of the shop's daily sales from the frequency table.
  1. Mean = Σfx ÷ Σf = 13,800 ÷ 40 = $345
  2. Median position = 40 ÷ 2 = 20th value; the cumulative frequencies 4, 14, 28 show it is in the class 300 – under 400
  3. Median = L + ((n/2 − F) ÷ f) × c = 300 + ((20 − 14) ÷ 14) × 100 = 300 + 42.86 = $342.86
  4. Modal class = 300 – under 400 (frequency 14); d₁ = 14 − 10 = 4 and d₂ = 14 − 8 = 6
  5. Mode = L + (d₁ ÷ (d₁ + d₂)) × c = 300 + (4 ÷ 10) × 100 = $340

Answer: Mean $345; median ≈ $342.86; mode ≈ $340

Characteristics, Advantages and Disadvantages

A good average should be easy to understand and calculate, based on all the values, not unduly affected by extreme values, clearly defined and suitable for further calculations. The table compares the three measures.

MeasureAdvantages and disadvantages
Mean Uses every value.It is used in further calculations such as standard deviation, but it is distorted by extreme values and may not be an actual value, such as 2.4 children.
Median The middle value.It is not affected by extreme values and suits skewed data such as wages, but it ignores the size of most values and is less useful for further calculations.
Mode The most common value.It is easy to find, is an actual value, and suits qualitative data such as the most popular shoe size, but there may be no mode or several modes.

2.6 Measures of Dispersion

Two sets of data can have the same average but be spread out very differently.

Measures of dispersion show how widely the values are scattered around the average.

Meaning and Types

Dispersion is the extent to which values differ from the average. The range is the highest value minus the lowest. The variance is the mean of the squared deviations from the mean, σ² = Σ(x − x̄)² ÷ n, or for a frequency distribution σ² = Σf(x − x̄)² ÷ Σf. The standard deviation is the square root of the variance, σ = √σ², in the same units as the data. The coefficient of variation is the standard deviation as a percentage of the mean, CV = σ ÷ x̄ × 100, used to compare the spread of data sets with different means.

Here you are measuring how spread out the figures are. Find the mean first, then how far each value is from it, square those distances and find their average: that is the variance. The standard deviation is the square root of the variance, and dividing it by the mean and multiplying by 100 gives the coefficient of variation. Below, we show you how it is done, step by step.

Ungrouped: the daily sales ($) of a kiosk over 5 days were 40, 50, 60, 70 and 80. Find the range, variance, standard deviation and coefficient of variation.
  1. Range = 80 − 40 = $40
  2. Mean = 300 ÷ 5 = $60
  3. Deviations: −20, −10, 0, 10, 20; squared: 400, 100, 0, 100, 400
  4. Variance = (400 + 100 + 0 + 100 + 400) ÷ 5 = 1,000 ÷ 5 = 200
  5. Standard deviation = √200 = $14.14
  6. CV = 14.14 ÷ 60 × 100 = 23.6%

Answer: Range $40; variance 200; standard deviation $14.14; CV 23.6%

Grouped: find the standard deviation of the shop's daily sales (mean $345).
  1. Deviations (x − 345): −195, −95, 5, 105, 205
  2. f(x − 345)²: 4 × 38,025 = 152,100; 10 × 9,025 = 90,250; 14 × 25 = 350; 8 × 11,025 = 88,200; 4 × 42,025 = 168,100
  3. Σf(x − x̄)² = 152,100 + 90,250 + 350 + 88,200 + 168,100 = 499,000
  4. Variance = 499,000 ÷ 40 = 12,475
  5. Standard deviation = √12,475 = $111.69
  6. CV = 111.69 ÷ 345 × 100 = 32.4%

Answer: Standard deviation ≈ $111.69; CV ≈ 32.4%

Salesperson A has mean weekly sales of $400 with a standard deviation of $40. Salesperson B has mean sales of $250 with a standard deviation of $50. Who is more consistent?
  1. CV for A = 40 ÷ 400 × 100 = 10%
  2. CV for B = 50 ÷ 250 × 100 = 20%
  3. The lower CV shows less spread compared with the mean

Answer: A is more consistent (CV 10% against 20%)

Characteristics, Advantages and Disadvantages

The table compares the main measures of dispersion. In general, a smaller standard deviation or CV means the data is more consistent, which for sales or returns usually means lower risk.

MeasureAdvantages and disadvantages
Range Highest minus lowest.It is very easy to calculate and understand, but it uses only two values and is distorted by one extreme value.
Variance Mean of squared deviations.It uses all values and is used in advanced statistics, but its units are squared, such as dollars squared, so it is hard to interpret.
Standard deviation Square root of variance.It uses all values, is in the same units as the data and is the most widely used measure, but it takes longer to calculate and is affected by extreme values.
Coefficient of variation Standard deviation as a percentage of the mean.It compares spread between data sets with different means or units, but it is not meaningful when the mean is close to zero.

All Financial Mathematics and Statistics 1 learning outcomes

  1. Learning Outcome 1: Business Computations
  2. Learning Outcome 2: Financial Statistics
  3. Learning Outcome 3: Time Value of Money